On the Complexity of Finding a Sparse Connected Spanning Subgraph in a Non-Uniform Failure Model

Authors Matthias Bentert, Jannik Schestag , Frank Sommer

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Author Details

Matthias Bentert
  • Department of Informatics, University of Bergen, Norway
Jannik Schestag
  • Faculteit Elektrotechniek, Wiskunde en Informatica, TU Delft, The Netherlands
  • Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, Germany
Frank Sommer
  • Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, Germany


This work was initiated at the research retreat of the Algorithmics and Computational Complexity group of TU Berlin held in Darlingerode in September 2022.

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Matthias Bentert, Jannik Schestag, and Frank Sommer. On the Complexity of Finding a Sparse Connected Spanning Subgraph in a Non-Uniform Failure Model. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We study a generalization of the classic Spanning Tree problem that allows for a non-uniform failure model. More precisely, edges are either safe or unsafe and we assume that failures only affect unsafe edges. In Unweighted Flexible Graph Connectivity we are given an undirected graph G = (V,E) in which the edge set E is partitioned into a set S of safe edges and a set U of unsafe edges and the task is to find a set T of at most k edges such that T - {u} is connected and spans V for any unsafe edge u ∈ T. Unweighted Flexible Graph Connectivity generalizes both Spanning Tree and Hamiltonian Cycle. We study Unweighted Flexible Graph Connectivity in terms of fixed-parameter tractability (FPT). We show an almost complete dichotomy on which parameters lead to fixed-parameter tractability and which lead to hardness. To this end, we obtain FPT-time algorithms with respect to the vertex deletion distance to cluster graphs and with respect to the treewidth. By exploiting the close relationship to Hamiltonian Cycle, we show that FPT-time algorithms for many smaller parameters are unlikely under standard parameterized complexity assumptions. Regarding problem-specific parameters, we observe that Unweighted Flexible Graph Connectivity admits an FPT-time algorithm when parameterized by the number of unsafe edges. Furthermore, we investigate a below-upper-bound parameter for the number of edges of a solution. We show that this parameter also leads to an FPT-time algorithm.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
  • Flexible graph connectivity
  • NP-hard problem
  • parameterized complexity
  • below-guarantee parameterization
  • treewidth


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